132 problems

For every $d$-dimensional Gorenstein polytope $P$ of index $r$, the stringy $E$-polynomial $E_{\mathrm{st}}(P;u,v)$ vanishes if and only if $P$ is thin, equivalently, its local…

For every convex polytope $P$ of dimension $d\ge 1$, let $L(P)$ be its face lattice and let $G(L(P))$ be its cover graph: the vertices are the faces of $P$, and two faces $F$ and…

For each positive integer $n$, define $g(n)$ to be the maximum number of facets of a full-dimensional polytope $P\subseteq\mathbb{R}^n$ whose vertices belong to $\{0,1\}^n$. Determ…

Mihail–Vazirani conjecture. Every $0/1$ polytope $P$ satisfies

Let $\Delta_{d-1}$ be the standard $(d-1)$-simplex, and let $n\Delta_{d-1}$ denote its dilation by $n$. A unit simplex is a simplex of normalized volume one in $n\Delta_{d-1}$, and…

Let $\mathcal{L}\in\mathcal{D}^k$ be a family of $k$ matroids on $V$, and let $w\in\mathbb{R}_{\ge0}^V$. Weighted fractional matching-cover conjecture. One has … This is presented…

Polynomial hitting-number conjecture. The quantity $h(M)$ is bounded by a polynomial in $r$.

Hirsch conjecture. If $\Delta$ is the boundary of a convex polytope, then $I_\Delta$ is Hirsch.

Non-revisiting Conjecture. There is a path from $u$ to $v$ which at every step enters a different facet of $P$.

Let $\mathcal{M}=(E,\mathcal{C})$ be an oriented matroid of rank $r$, let $e\in E$, and let $\mathcal{C}^+$ denote its positive circuits. An $e$-positive circuit is a positive circ…

3-geodesic conjecture. For every $d$-dimensional Dantzig figure $(P,x,y)$,

Strong Dantzig conjecture. Every fundamental deformation of a $d$-dimensional Dantzig figure is good; equivalently,

Dantzig conjecture. The graph $E(d)$ is strongly connected: for any vertices $x,y$ of $E(d)$, there is an oriented path from $x$ to $y$.

Face-structure conjectures for tropical polytopes. The following assertions hold: (1) $k$-faces of tropical polytopes are extreme sets; (2) the topological boundary of a $k$-face i…

Tropical halfspace characterization conjecture. A tropical polytope $P$ is pure and full dimensional if and only if it has a halfspace description whose apices are in general posit…

Tropical halfspace representation conjecture. If $P$ is pure, then the halfspaces from a generic lift $\overline{P}$ of $P$ map to tropical halfspaces whose intersection is $P$ its…

Combinatorial characterization conjecture. If the assignment $M\mapsto G_M$ satisfies 1. for every such decomposition,

Let $\mathrm{met}_n$ be the metric polytope, and let its fractional vertices be the vertices that are not integral. The restriction of $\mathrm{met}_n$ to its fractional vertices i…

Partition characterization. Every minimal halfspace with respect to ${\mathcal P}$ has the form

A tropical $d$-plane in $n$-space is a tropical linear space of dimension $d$ in $n$-space. Let $i$ be a face dimension, and map the space to the quotient by the diagonal line: … A…

Collapsing conjecture. Suppose that

Let $i$ be the number of faces of an irreducible $i$-hedrite, and let a central circuit be a central circuit of the hedrite. An irreducible $i$-hedrite is maximal irreducible when…

Oriented-multicut adjacency conjecture. (i) An oriented multicut $\delta'(A)$ is not adjacent to every oriented multicut $\delta'(B)$ such that $B<A^T$. (ii) The orbit represented…

Oriented-multicut facet conjecture. The following inequalities define facets of $OMCUT_n$: (i) zero-extensions of every facet of $OMCUT_{n-1}$, including oriented triangle and non-…

Oriented-multicut skeleton conjecture. (i) All oriented multicuts are extreme rays of $QMET_n$. (ii) The oriented cuts form a dominating clique in $G_{OMCUT_n}$; consequently, the…